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PropagationHalfLength ¤

Half-length model for the decay of the propagation rate coefficient with chain length.

This model implements the chain-length dependence:

\[ k_p(i) = k_p \left[ 1+ (C - 1)\exp{\left (-\frac{\ln 2}{i_{1/2}} (i-1) \right )} \right] \]

where \(k_p=k_p(\infty)\) is the long-chain value of the propagation rate coefficient, \(C\ge 1\) is the ratio \(k_p(1)/k_p\) and \((i_{1/2}+1)\) is the hypothetical chain-length at which the difference \(k_p(1) - k_p\) is halved.

References

  • Smith, Gregory B., et al. "The effects of chain length dependent propagation and termination on the kinetics of free-radical polymerization at low chain lengths." European polymer journal 41.2 (2005): 225-230.
PARAMETER DESCRIPTION
kp

Long-chain value of the propagation rate coefficient, \(k_p\).

TYPE: Arrhenius | Eyring

C

Ratio of the propagation coefficients of a monomeric radical and a long-chain radical, \(C\).

TYPE: float DEFAULT: 10.0

ihalf

Half-length, \(i_{1/2}\).

TYPE: float DEFAULT: 1.0

name

Name.

TYPE: str DEFAULT: ''

Examples:

>>> from polykin.kinetics import PropagationHalfLength, Arrhenius
>>> kp = Arrhenius(
...    10**7.63, 32.5e3/8.314, Tmin=261., Tmax=366.,
...    symbol='k_p', unit='L/mol/s', name='kp of styrene')

>>> kpi = PropagationHalfLength(kp, C=10, ihalf=0.5,
...    name='kp(T,i) of styrene')
>>> kpi
name:    kp(T,i) of styrene
C:       10
ihalf:   0.5
kp:
  name:            kp of styrene
  symbol:          k_p
  unit:            L/mol/s
  Trange [K]:      (261.0, 366.0)
  k0 [L/mol/s]:    42657951.88015926
  EaR [K]:         3909.0690401732018
  T0 [K]:          inf

>>> kpi(T=50., i=3, Tunit='C')
371.75986615653215
Source code in src/polykin/kinetics/cldpropagation.py 
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class PropagationHalfLength(KineticCoefficientCLD):
    r"""Half-length model for the decay of the propagation rate coefficient
    with chain length.

    This model implements the chain-length dependence:

    $$ k_p(i) = k_p \left[ 1+ (C - 1)\exp{\left (-\frac{\ln 2}{i_{1/2}} (i-1)
                \right )} \right] $$

    where $k_p=k_p(\infty)$ is the long-chain value of the propagation rate
    coefficient, $C\ge 1$ is the ratio $k_p(1)/k_p$ and $(i_{1/2}+1)$ is the
    hypothetical chain-length at which the difference $k_p(1) - k_p$ is halved.

    **References**

    *   Smith, Gregory B., et al. "The effects of chain length dependent
        propagation and termination on the kinetics of free-radical
        polymerization at low chain lengths." European polymer journal
        41.2 (2005): 225-230.

    Parameters
    ----------
    kp : Arrhenius | Eyring
        Long-chain value of the propagation rate coefficient, $k_p$.
    C : float
        Ratio of the propagation coefficients of a monomeric radical and a
        long-chain radical, $C$.
    ihalf : float
        Half-length, $i_{1/2}$.
    name : str
        Name.

    Examples
    --------
    >>> from polykin.kinetics import PropagationHalfLength, Arrhenius
    >>> kp = Arrhenius(
    ...    10**7.63, 32.5e3/8.314, Tmin=261., Tmax=366.,
    ...    symbol='k_p', unit='L/mol/s', name='kp of styrene')

    >>> kpi = PropagationHalfLength(kp, C=10, ihalf=0.5,
    ...    name='kp(T,i) of styrene')
    >>> kpi
    name:    kp(T,i) of styrene
    C:       10
    ihalf:   0.5
    kp:
      name:            kp of styrene
      symbol:          k_p
      unit:            L/mol/s
      Trange [K]:      (261.0, 366.0)
      k0 [L/mol/s]:    42657951.88015926
      EaR [K]:         3909.0690401732018
      T0 [K]:          inf

    >>> kpi(T=50., i=3, Tunit='C')
    371.75986615653215
    """

    kp: Union[Arrhenius, Eyring]
    C: float
    ihalf: float
    symbol: str = 'k_p(i)'

    def __init__(self,
                 kp: Union[Arrhenius, Eyring],
                 C: float = 10.0,
                 ihalf: float = 1.0,
                 name: str = ''
                 ) -> None:

        check_type(kp, (Arrhenius, Eyring), 'kp')
        check_bounds(C, 1., 100., 'C')
        check_bounds(ihalf, 0.1, 10, 'ihalf')

        self.kp = kp
        self.C = C
        self.ihalf = ihalf
        self.name = name

    def __repr__(self) -> str:
        return custom_repr(self, ('name', 'C', 'ihalf', 'kp'))

    @staticmethod
    def equation(i: Union[int, IntArray],
                 kp: Union[float, FloatArray],
                 C: float,
                 ihalf: float
                 ) -> Union[float, FloatArray]:
        r"""Half-length model chain-length dependence equation.

        Parameters
        ----------
        i : int | IntArray
            Chain length of radical.
        kp : float | FloatArray
            Long-chain value of the propagation rate coefficient, $k_p$.
        C : float
            Ratio of the propagation coefficients of a monomeric radical and a
            long-chain radical, $C$.
        ihalf : float
            Half-length, $i_{i/2}$.

        Returns
        -------
        float | FloatArray
            Coefficient value.
        """

        return kp*(1 + (C - 1)*exp(-log(2)*(i - 1)/ihalf))

    def __call__(self,
                 T: Union[float, FloatArrayLike],
                 i: Union[int, IntArrayLike],
                 Tunit: Literal['C', 'K'] = 'K'
                 ) -> Union[float, FloatArray]:
        r"""Evaluate kinetic coefficient at given conditions, including unit
        conversion and range check.

        Parameters
        ----------
        T : float | FloatArrayLike
            Temperature.
            Unit = `Tunit`.
        i : int | IntArrayLike
            Chain length of radical.
        Tunit : Literal['C', 'K']
            Temperature unit.

        Returns
        -------
        float | FloatArray
            Coefficient value.
        """
        TK = convert_check_temperature(T, Tunit, self.kp.Trange)
        if isinstance(i, (list, tuple)):
            i = np.array(i, dtype=np.int32)
        return self.equation(i, self.kp(TK), self.C, self.ihalf)

__call__ ¤

__call__(
    T: Union[float, FloatArrayLike],
    i: Union[int, IntArrayLike],
    Tunit: Literal["C", "K"] = "K",
) -> Union[float, FloatArray]

Evaluate kinetic coefficient at given conditions, including unit conversion and range check.

PARAMETER DESCRIPTION
T

Temperature. Unit = Tunit.

TYPE: float | FloatArrayLike

i

Chain length of radical.

TYPE: int | IntArrayLike

Tunit

Temperature unit.

TYPE: Literal['C', 'K'] DEFAULT: 'K'

RETURNS DESCRIPTION
float | FloatArray

Coefficient value.
Source code in src/polykin/kinetics/cldpropagation.py 
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def __call__(self,
             T: Union[float, FloatArrayLike],
             i: Union[int, IntArrayLike],
             Tunit: Literal['C', 'K'] = 'K'
             ) -> Union[float, FloatArray]:
    r"""Evaluate kinetic coefficient at given conditions, including unit
    conversion and range check.

    Parameters
    ----------
    T : float | FloatArrayLike
        Temperature.
        Unit = `Tunit`.
    i : int | IntArrayLike
        Chain length of radical.
    Tunit : Literal['C', 'K']
        Temperature unit.

    Returns
    -------
    float | FloatArray
        Coefficient value.
    """
    TK = convert_check_temperature(T, Tunit, self.kp.Trange)
    if isinstance(i, (list, tuple)):
        i = np.array(i, dtype=np.int32)
    return self.equation(i, self.kp(TK), self.C, self.ihalf)

equation staticmethod ¤

equation(
    i: Union[int, IntArray],
    kp: Union[float, FloatArray],
    C: float,
    ihalf: float,
) -> Union[float, FloatArray]

Half-length model chain-length dependence equation.

PARAMETER DESCRIPTION
i

Chain length of radical.

TYPE: int | IntArray

kp

Long-chain value of the propagation rate coefficient, \(k_p\).

TYPE: float | FloatArray

C

Ratio of the propagation coefficients of a monomeric radical and a long-chain radical, \(C\).

TYPE: float

ihalf

Half-length, \(i_{i/2}\).

TYPE: float

RETURNS DESCRIPTION
float | FloatArray

Coefficient value.
Source code in src/polykin/kinetics/cldpropagation.py 
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@staticmethod
def equation(i: Union[int, IntArray],
             kp: Union[float, FloatArray],
             C: float,
             ihalf: float
             ) -> Union[float, FloatArray]:
    r"""Half-length model chain-length dependence equation.

    Parameters
    ----------
    i : int | IntArray
        Chain length of radical.
    kp : float | FloatArray
        Long-chain value of the propagation rate coefficient, $k_p$.
    C : float
        Ratio of the propagation coefficients of a monomeric radical and a
        long-chain radical, $C$.
    ihalf : float
        Half-length, $i_{i/2}$.

    Returns
    -------
    float | FloatArray
        Coefficient value.
    """

    return kp*(1 + (C - 1)*exp(-log(2)*(i - 1)/ihalf))